University Statistics Course: Exercise 4 Assignment Analysis

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Added on 2022/08/23

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This document contains the solutions to a statistics assignment (Exercise 4) focusing on core concepts in probability and statistical inference. The assignment covers the definition and application of sampling distributions, the central limit theorem (CLT), and the interpretation of confidence intervals. It also includes problems on mutually exclusive events and their probabilities, coin flip trials, and the law of large numbers. Furthermore, the solution addresses a confidence interval calculation and interpretation, along with a Chi-square test, including null hypothesis formulation, computation of expected and observed frequencies, and interpretation of results. The document provides a comprehensive breakdown of each question, demonstrating the application of statistical principles to real-world scenarios.

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Exercise 4
Instructor:
Name:
Date:
Exercise 4
1. What is a “sampling distribution?”
A sampling distribution is a probability distribution of a statistic obtained through a large
number of samples drawn from a specific population. For instance, in a sensitization conference
hosting both men and women, a person may be interested in knowing the number of males and
females that attendant the meeting. The above variable (gender) can be used to create the sample
distribution of ethics in the city. Standard error exhibits the expected dispersion from sample to
sample.
2. What is the central limit theorem? Lebron James
If Lebron James was not only a professional basketball player but also a super model,
professor, or had any other title then he could have a significant impact not only in the country
by also across the globe. Similarly, central limit theory plays a critical role in statistics especially
in activities that involve using a sample to make conclusions or inferences about a population.
3. What does a 99% confidence interval around a sample mean tell you versus a 95%
confidence interval around
One of the measures use to compute the confidence interval is the z value thus the higher
the value the greater the interval. Notably, the z value of 99% is 2.58 whereas that of 95% is
1.96. Therefore, at 99% the confidence interval of the mean value is higher than that of 95%.

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5. Confidence Interval
CI =μ ±
z α
2
∗σ
√ n =28 ± 1.96∗3.5
√ 200 =28 ± 1.96∗0.2475
¿ 28 ± 0.49=27.51∧28.49
The results above exhibit that average amount given to the Bernie Sanders Campaign
falls within $27.51 and $28.49
6. In dealing with probability, what does it mean to say the outcome of events are
“mutually exclusive?” Give an example of events that are and are not mutually
exclusive.
Events are said to be mutually exclusive if the occurrence of any one of them means the
others will not occur. For instance, while a fair tossing a coin, one can only get one outcome,
either head or tail. On the other side not mutually, exclusive events can occur without affecting
the other. For instance, in playing cards one has a card that has both king and heart.
7. Suppose you flip a coin three times.
a. Are the trials independent or conditional probabilities?
The trials are independent since one outcome does not affect the other outcome.
b. How many total outcomes are possible?
There are 2 outcomes (head or tail) and 3 trails thus 23=8 possible outcomes
c. What is the probability of each unique possible outcome?
There are 8 unique possible outcomes thus the probability for each unique possible
outcome is given as 1
8 or 0.125

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d. What are the possible outcomes for the total number of heads out of three tosses, and
what are their probabilities?
There are 3 possible outcomes for the total number of heads out the three counts. The
probability is given as 0.125
6. Biasness of a coin
∑ nCx px ( 1− p )n−x=5 C 5 0.55 ( 1−0.5 )5−5 +5 C 4 0.54 ( 1−0.5 )5−4
+5 C 3 0.53 ( 1−0.5 ) 5−3 +5 C 2 0.52 ( 1−0.5 ) 5−2+5 C 1 0.51 ( 1−0.5 ) 5−1
+5 C 0 0.50 ( 1−0.5 )5−0=1
The coin is biased since the probability of getting five heads is 1.
7. The law of large numbers
The law of large numbers exhibits that as the number of trials increases, the average of
the outcomes will get closer and closer to its expected value. Similarly, the probabilities
associated with all casino games favor the house (assuming that the casino can successfully
prevent blackjack players from counting cards) thus casinos always make money in the long run.
8. Monte Hall problem
Notably, by switching after a door is opened, I get the benefit of choosing two doors
rather thus than one thus my probability increases from 1/3 to 2/3
9. Binomial
a) Exactly five heads
nCx px ( 1− p ) n−x=11C 5 0.55 ( 1−0.5 ) 11−5=0.2256
b) At least five heads
∑ nCx px ( 1− p )n−x=11C 0 0.50 ( 1−0.5 )11−0 +11C 4 0.54 ( 1−0.5 )11−4

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+11C 3 0.53 ( 1−0.5 )11−3 +11C 2 0.52 ( 1−0.5 )11−2+11C 1 0.51 ( 1−0.5 )11−1
¿ 0.5
1 – 0.5=0.5
c) How many heads
At 11 flips there are expected 5 heads
10. Chi Square Problem
I. Purpose of Chi square test
It is used as a inferential statistic
II. Null hypothesis
There is no association between gender and part affiliation
III. Compute the proper percentages
Observed Frequencies
Category Male Female Colum Total
Republican
0.85714
3 0.142857 1
Democrat
0.66666
7 0.333333 1
IV. Critical region
X0.05 ,1=3.841
V. Level of measurement
Nominal level
VI. Compute the probabilities
a) 0.8571
b) 0.1429
c) 0.6667
d) 0.3333

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VII. Expected frequencies
a) 52
b) 18
c) 88
d) 32
VIII.
Null Table
Category Male Female Colum Total
Republican 52 18 70
Democrat 88 32 120
Row Total 140 50 190
IX. Chi square
X2= ( O−E ) 2
E =1.18+7.09+0.89+1.77=10.93
Since the chi square calculated 10.93 is greater than the critical value the null
hypothesis will be rejected.
10. Chi square problem
a) Null hypothesis
There is no association between party identification and race
b) Critical region
X0.05 ,2=9.21
c) Chi square
X2= ( O−E )2
E =13.24
d) Interpret the results